A complex manifold is said to be complex hyperbolic if it does not contain
the image of any non-trivial holomorphic map from the complex line.
Diophantine geometry studies questions about finiteness of rational
points on algebraic manifolds. The two subjects are highly related. In
this talk we examine the similarity in formulation and key steps.
At the same time, we will try to point out the differences in the
techniques available. For complex hyperbolicity, we would explain recent results in
Kobayashi Conjecture and Lang Conjecture. For diophantine approximation,
we would mention some approaches to smooth complex two ball quotients, which
are the quotients of the unit ball in the two dimensional complex plane by
cocompact lattices.